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Give a formula for f^(n) (x) [**the nth derivative] valid for all n >= 1 and find an upper bound for | f^(n) (x) | on the interval -0.25 <= x <= 0.25.

[ the error ].

I found the nth derivative to be

f^(n) (x) = (-1)^n+1 * 2^n /n * n!

--------------------------------

(1 + 2x)^n

so for

first derivative = 2 / 1+2x

second " " = -4 / (1+2x)^2

third " " = 16 / (1+2x)^3

etc.

now for the error i kno there is a lagrange error bound equation for taylor polynomials, but the question isnt for the taylor polynomial, only the "derivative generator"

i kno the max |f^(n+1)| <= M on an interval

so i just need help dealing with only the derivative error and i also want to kno how to find the M value in general with Taylor polynomials (not part of the above question)

where |f(x) - P_n(x)| <= M / (n+1)! * |x-a|^(n+1)

for interval between a and x